(p.2 ( ) 1 2 ( ) Fisher, Ronald A.1932, 1971, 1973a, 1973b) treatment group controll group (error function) 2 (Legendre, Adrian

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1 Maddala(1993) Mátyás and Sevestre (1996) Hsiao(2003) Baltagi(2001) Lee(2002) Woolridge(2002a), Arellano(2003) Journal of Econometrics Econometrica Greene(2000) Maddala(2001) Johnston and Di- Nardo(1997) Woolridge(2000a) Handbook of Econometrics (North-Holland) Chamberlain (1984) Arellano and Honoré (2001) Gauss, Carl Friedrich (Airy, George Biddell) (Poincaré, Henri)

2 (p.2 ( ) 1 2 ( ) Fisher, Ronald A.1932, 1971, 1973a, 1973b) treatment group controll group (error function) 2 (Legendre, Adrian Marie: ) 1805) R.A. 1

3 (erro components model) heterogeneous y it = α + X itβ + u it i = 1,...N; t = 1,...T (1) i two-way error component 5 u it = µ i + λ t + ν it (2) µ i λ t ν it λ t = 0 (2) (one-way error component) u it = µ i + ν it (3) 5 n

4 (1) OLS (pooling estimation) one-way fixed effect estimation (Least Squares Dummy Variable Model ; LSDV) (one-way random effect estimation) y it θy i X it θx i (Generalized Least Squares; GLS) θ Baltagi(2001) Hsiao(2002)

5 (self selectivity) (non-response) (attrition) Panel Study of Income Dynamics (PSID) National Longitudinal Surveys of Labor Market Experience (NLS) 5 ( ) (5020 ) ( ) ( ) ( ) (5159 ) ( ) NLSY NLSY (Statistics Canada) The Canadian Survey of Labor Income Dynamics (SLID) The German Social Economic Panel The Belgian Socioeconomic Panel The French Household Panel

6 The Hungarian Household Panel ( ) The British Household Panel Survey (BHPS) The Dutch Socio-Economic Panel (ISEP) The Russian Longitudinal Monitoring Survey (RLMS) The Swiss Household Panel (SHP) The Luxembourg Panel Socio-Economique Liewen zu L zebuerg (PSELL) EuroStat 1994 The European Community Household Panel (ECHP) ECHP 1.3 Bayes, Thomas (Savage, Leonard J) 6 Bayes theorem A H 1, H 2,... H k A H i P (H i A) P (A H i ) P (H i A) H 1, H 2,... H k H i H 2... H k = Ω P (H i A) = P (H i ) P (A H i ) sum{p (H j ) P (A H j )} P (H i ) H i prior probability P (H i A) posterior probability

7 (Zellner, Arnold) (Ordinary Least Squares; OLS) Y 1, Y 2... Y n µ Y var(y ) = σ 2 Y Y i = µ Y + ε i E(ε i ) = 0, var(ε i ) = σ 2 Y E(Y i ) = µ Y e i = Y i µ Y i = 1,..., n µ Y min n e 2 i = min n (y i µ Y ) 2 i=1 (sum of squared errors, SSE) µ Y SSE µ Y = d n (y i µ Y ) 2 i=1 i=1 = 2 n (Y i ˆµ Y ) = 0 dµ Y i=1 n Y i = nˆµ Y ˆµ Y = i=1 n Y i i=1 ˆµ Y (Best Linear Unbiased Estimator; BLUE) Gauss-Markov s theorem n 7 Koop(2003) Lancaster(2004)

8 y = β 0 + β 1 x + u (4) x u Cov(x, u) 0 (5) β 0 β 1 x x u z Cov(z, u) = 0 (6) Cov(z, x) 0 (7) x Cov(z, u) = 0 Cov(z, x) 0 1 = 0 x = π 0 + π 1 z + v (8) π 1 = Cov(z, x)/v ar(z) (7) π 1 0 ( ) ˆβ 1 = n (z i z)(y i ȳ) i=1 n (z i z) (x i x) i=1 (9) 8 Bowden and Turkington (1984) Wooldridge (2003) 15 Wooldridge(2003) 9 Philip G. Wright(1928) B Stock and Trebbi (2003) Philip Wright Sewall Wright Philip Wright

9 ˆβ 0 = ȳ ˆβ 1 x (10) z = x (6) (7) p lim( ˆβ 1 ) = β 1 (6) (7) x u 10 z x z u p lim ˆβ 1 = β 1 + Corr(z, u) Corr(z, x) σu σ x (11) σ u σ x u x Corr(z, u) Corr(z, x) ˆβ 1 11 y 2 y 1 = β 0 + β 1 y 2 + β 2 z 1 + β 3 z 2 + u 1 (12) z 1 z 2 z 3 z 4 Hausman (1978) y 2 12 y 2 y 2 = α 0 + α 1 z 1 + α 2 z 2 + α 3 z 3 + α 4 z 4 + v 2 (13) z j u 1 v 2 u 1 y 2 u 1 1 = 0 y 2 u Weak Instrumental Variables Staiger and Stock (1997) Nelson and Sartz (1990) Gary Chamberlain Jerry Hausman Christopher Sims 12 Durbin-Wu-Hausman test Wu-Hausman test Bowden and Turkington (1984, pp.50-52) Davidson and MacKinnon (2004, pp )

10 u 1 = δ 1 v 2 + e 1 (14) 13 ˆv 12 y 1 = β 0 + β 1 y 2 + β 2 z 1 + β 3 z 2 + δ 1ˆv 2 + ε (15) t 1 = 0 y 2 (l) k l k (4) 6 (1) (4) û (2) û (l) R 2 (3) û nr 2 χ 2 (l k) (16) (Method of Moments) (Pearson, Karl) f(x θ) θ

11 E(x) = xf(x θ)dx = g(θ) (17) E(x) θ θ θ = g 1 (E(x)) (18) E(x) x ˆθ = g 1 ( x) (19) ˆθ θ 13 (Generalized Method of Moments; GMM) 14 (overidentified) 15 p q 2 Q(θ) arg min θ Q(θ) (20) Q(θ) = f(θ) Af(θ) A (weighting matrix) Q(θ) 0 f(θ) = 0 Q(θ) = 0 E(f(θ)) = 0 q f(θ) Hansen(1982) Hansen and Singleton(1982) 15 (Hansen(1982))

12 (Edgeworth, Francis Ysidro) 16 5 y = (y 1, y n ) θ = (θ 1, θ 2, θ p ) likelihood L(θ) 17 L(θ) θ θ maximum likelihood estimator θ L(θ) logl(θ) θ log(lθ) θ = 0 (21) y i = βx i + ε i i = 1,, n (22) ε i N(0, σ 2 ) 16 Pratt (1976) 1997 pp Silvey (1970) Cox and Hinkley (1974) (1995)

13 { log L(β) = log [(2πσ 2 ) n2 exp (y }] βx) (y βx) 2σ 2 = n 2 log(2πσ2 ) 1 2σ 2 (y βx) (y βx) β (23) β = ˆβ = x y/x x (24) log L(β) 2 log(lβ) β 2 = x x/σ 2 (25) β 2 log L(β) β I(θ) y y f θ (y) { 2 } log L(θ) I(θ) = E θ 2 { 2 } log f θ (y) = E θ 2 (26) I(θ) V {t(y)} 1 I(θ) (27) θ Z H 0 : θ = θ 0 n( θ θ 0 ) N(0, 1/I 1 (θ 0 )) θ 1 > θ 0 ni1 (θ 0 )( θ θ 0 ) > Z α (28) I 1 (θ) = I(θ)/n Z α α Z

14 H 0 : θ = θ 0 H 1 : θ θ 0 H 0 θ 0 χ 2 (1) 2 log L( θ) L(θ 0 ) > χ2 α(1)(= Z 2 α/2 ) (29) χ 2 α(1) = Z 2 α/2 (Wald Tests) r r(θ) = 0 r(θ) V ar(r(ˆθ)) R(θ 0 )V ar(ˆθ)r (θ 0 ) (30) R(θ) r i (θ)/ θ i r k V ar(ˆθ) W = r (ˆθ)(R(ˆθ) V ar(ˆθ)r (ˆθ)) 1 r(ˆθ) (31) H 0 : θ = θ 0 χ 2 (r) (Lagrange Multiplier Tests) l(θ) r(θ) = 0 θ l(θ) r (θ)λ (32) g( θ) R ( θ) λ = 0 (33) r( θ) = 0 λ LM = λ R( θ)ĩ 1 R ( θ) λ (34) H 0 : θ = θ 0 χ 2 (r) Davidson and MacKinnon (2004, 10

15 µ 1, µ 2,... µ a (a 3) analysis of variance: ANOVA (factor) (level (treatment 20 A A 1,..., A a r 1,..., r a A i j y ij y ij = µ i + ε ij i = 1, 2,..., a; j = 1,..., r i (35) µ i i ε ij N(0, σ 2 ) n = Σr i r i µ i µ = r i µ i n grand mean completely randomized design (randomized block design) 1969

16 (effect) α i = µ i µ r i α i = 0 (30) y ij = µ + α i + ε ij i = 1, 2,... ; j = 1,..., r i (36) ( µ)+ i α i + ε ij ) one-way layout A H 0 : µ 1 = µ 2 =... µ a H 0 : α 1 = α 2 =... α a = 0 S e = (y ij ȳ i ) 2 i j = yij 2 yi 2 r i (37) i j i S e σ 2 ν e = n a H 0 : µ 1 = µ 2 =... µ a y ij = µ + ε ij µ ȳ S T = i H 0 (y ij ȳ i ) 2 = yij 2 ȳi 2 n (38) j i j S A = S T S e = yi 2 r i y 2 n i = r i (ȳ i ȳ) 2 (39) i S A S e ν A = a 1 F = S A ν A S e ν e (40) ν A ν e F (ν A, ν e ) (ANOVA Test) (35) (S A ) (S A ν A ) (S e ) (S e ν e )

17 V i = (y ij ȳ i ) 2 (r i 1) i = 1,..., a i V e = (r i 1)V i (n a) = c i (y ij ȳ i ) 2 (n a) j B = (n a) log V e (r i 1) log V i (41) i H 0 : σ1 2 = σ2 2 =... = σa 2 a 1 B = (a 1) B { i 1 r 1 i 1 n a } (42) F 2 ( main effect () (one factor at a time experiment) ( (factorial experiment) ) A B ,pp ) 22

18 A i B j k y ijk AB a, b r y ijk = µ ij + e ijk i = 1, 2,... a; j = 1,..., b; k = 1, 2,..., r (43) µ ij A i B j e ijk N(0, σ 2 ) n = abr A i A i B j B j µ i = µ ij b, µ j = µ ij a, µ j i i µ ij ab (44) µ A i B j µ main effect α i = µ i µ, i = 1,... a; β j = µ j µ, j = 1,... b (45) µ ij A, B (interaction) (αβ) ij = µ ij (µ + α i + β j ) = µ ij µ i µ j + µ i = 1, 2,... a; j = 1, 2..., b (46) (38) (41) y ijk = µ + α i + β j + (αβ) ij + ε ijk (47) ()+( A )+( B )+( AB )+( ) Σα i = 0, Σβ j = 0, (αβ) ij = 0, j = 1,... b; (αβ) ij = 0, i = i j 1,... a S A, S B, S AB S e j v T = n 1, v A = a 1, v B = b 1, v AB = (a 1)(b 1), v e = ab(r 1) (48) V A = S A v A, V B = S B v B, V AB = S AB v AB, V e = S e v e (49) F H 0 : (αβ) ij 0 F AB = V AB V e H 0 : α i 0 F A = V A V e H 0 : β j 0 F B = V B V e

19 y ij = µ + α i + β j + (αβ) ij + ε ij (50) S e

4.9 Hausman Test Time Fixed Effects Model vs Time Random Effects Model Two-way Fixed Effects Model

4.9 Hausman Test Time Fixed Effects Model vs Time Random Effects Model Two-way Fixed Effects Model 1 EViews 5 2007 7 11 2010 5 17 1 ( ) 3 1.1........................................... 4 1.2................................... 9 2 11 3 14 3.1 Pooled OLS.............................................. 14